Solve 1/x^2+3x-4+1/x^2+2x-8=1/x^2-8x+12

$$\frac { 1 } { x ^ { 2 } + 3 x - 4 } + \frac { 1 } { x ^ { 2 } + 2 x - 8 } = \frac { 1 } { x ^ { 2 } - 8 x + 12 }$$

Solve for x

$$x=\sqrt{59}+9\approx 16.681145748$$

$$x=9-\sqrt{59}\approx 1.318854252$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,1,2,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x-2\right)\left(x-1\right)\left(x+4\right), the least common multiple of x^{2}+3x-4,x^{2}+2x-8,x^{2}-8x+12.

$$\left(x-6\right)\left(x-2\right)+\left(x-6\right)\left(x-1\right)=\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply x-6 by x-2 and combine like terms.

$$x^{2}-8x+12+\left(x-6\right)\left(x-1\right)=\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply x-6 by x-1 and combine like terms.

$$x^{2}-8x+12+x^{2}-7x+6=\left(x-1\right)\left(x+4\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-8x+12-7x+6=\left(x-1\right)\left(x+4\right)$$

Combine -8x and -7x to get -15x.

$$2x^{2}-15x+12+6=\left(x-1\right)\left(x+4\right)$$

Add 12 and 6 to get 18.

$$2x^{2}-15x+18=\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$2x^{2}-15x+18=x^{2}+3x-4$$

Subtract x^{2} from both sides.

$$2x^{2}-15x+18-x^{2}=3x-4$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-15x+18=3x-4$$

Subtract 3x from both sides.

$$x^{2}-15x+18-3x=-4$$

Combine -15x and -3x to get -18x.

$$x^{2}-18x+18=-4$$

Add 4 to both sides.

$$x^{2}-18x+18+4=0$$

Add 18 and 4 to get 22.

$$x^{2}-18x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 22}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 22}}{2}$$

Multiply -4 times 22.

$$x=\frac{-\left(-18\right)±\sqrt{324-88}}{2}$$

Add 324 to -88.

$$x=\frac{-\left(-18\right)±\sqrt{236}}{2}$$

Take the square root of 236.

$$x=\frac{-\left(-18\right)±2\sqrt{59}}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{59}}{2}$$

Now solve the equation x=\frac{18±2\sqrt{59}}{2} when ± is plus. Add 18 to 2\sqrt{59}.

$$x=\frac{2\sqrt{59}+18}{2}$$

Divide 18+2\sqrt{59} by 2.

$$x=\sqrt{59}+9$$

Now solve the equation x=\frac{18±2\sqrt{59}}{2} when ± is minus. Subtract 2\sqrt{59} from 18.

$$x=\frac{18-2\sqrt{59}}{2}$$

Divide 18-2\sqrt{59} by 2.

$$x=9-\sqrt{59}$$

The equation is now solved.

$$x=\sqrt{59}+9 x=9-\sqrt{59}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,1,2,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x-2\right)\left(x-1\right)\left(x+4\right), the least common multiple of x^{2}+3x-4,x^{2}+2x-8,x^{2}-8x+12.

$$\left(x-6\right)\left(x-2\right)+\left(x-6\right)\left(x-1\right)=\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply x-6 by x-2 and combine like terms.

$$x^{2}-8x+12+\left(x-6\right)\left(x-1\right)=\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply x-6 by x-1 and combine like terms.

$$x^{2}-8x+12+x^{2}-7x+6=\left(x-1\right)\left(x+4\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-8x+12-7x+6=\left(x-1\right)\left(x+4\right)$$

Combine -8x and -7x to get -15x.

$$2x^{2}-15x+12+6=\left(x-1\right)\left(x+4\right)$$

Add 12 and 6 to get 18.

$$2x^{2}-15x+18=\left(x-1\right)\left(x+4\right)$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$2x^{2}-15x+18=x^{2}+3x-4$$

Subtract x^{2} from both sides.

$$2x^{2}-15x+18-x^{2}=3x-4$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-15x+18=3x-4$$

Subtract 3x from both sides.

$$x^{2}-15x+18-3x=-4$$

Combine -15x and -3x to get -18x.

$$x^{2}-18x+18=-4$$

Subtract 18 from both sides.

$$x^{2}-18x=-4-18$$

Subtract 18 from -4 to get -22.

$$x^{2}-18x=-22$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-22+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-22+81$$

Add -22 to 81.

$$x^{2}-18x+81=59$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=59$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{59}$$

Simplify.

$$x-9=\sqrt{59} x-9=-\sqrt{59}$$

Add 9 to both sides of the equation.

$$x=\sqrt{59}+9 x=9-\sqrt{59}$$